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Suppose $f$ and $g$ are continuous real valued functions on $[a, b]$ and are differentiable on $(a, b)$. Assume that $g^{\prime}(x) \neq 0$ for any $x \in(a, b)$. Prove that there exists $\xi \in(a, b)$ such that $$ \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)} $$
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